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Unformatted text preview: ECS 120: Theory of Computation Homework 2 Due: Oct. 12, beginning of class or noon in 2131 Kemper Problem 1. Show that if M = ( Q, Σ , δ, q , F ) is a DFA accepting the language L then M = ( Q, Σ , δ, q , Q − F ) is a DFA accepting the complement of L , L . Solution: Let M and M be as stated above, and L = L ( M ). Then, by definition, L = { w 1  ˆ δ ( q , w 1 ) ∈ F } . By definition, L , the complement of L , contains all words over Σ that are not in L , or all words not accepted by M , i.e. L = { w 2  ˆ δ ( q , w 2 ) 6∈ F } . This is equivalent to L = { w 2  ˆ δ ( q , w 2 ) ∈ Q \ F } , which, by definition, is accepted by M , i.e. L = L ( M ). Problem 2. (a) Show that there is an ( n + 2)state NFA for L n = (Σ ∗ )0Σ n . (Take Σ = { , 1 } .) Just draw it! (b) Prove that any DFA for L n requires at least 2 n states. Fix n . Suppose to the contrary that there exists a DFA M = ( Q, Σ , δ, q , F ) for L n that has fewer than 2 n states. Consider all possible strings x 1 , . . . , x 2 n of Σ n (all 2 n of them). By the pigeonhole principle, some two of these strings, x i and x j , where x i 6 = x j , must “collide” when processed by M n ; that is, ˆ δ ( q , x i ) = ˆ δ ( q , x j )....
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This note was uploaded on 05/06/2008 for the course ECS 120 taught by Professor Filkov during the Fall '07 term at UC Davis.
 Fall '07
 Filkov

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